Theorem clnbgrssvtx 49037

Description: The closed neighborhood of a vertex 𝐾 in a graph is a subset of all vertices of the graph. (Contributed by AV, 9-May-2025.)

Hypothesis

Ref	Expression
clnbgrvtxel.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
clnbgrssvtx	⊢ (𝐺 ClNeighbVtx 𝐾) ⊆ 𝑉

Proof of Theorem clnbgrssvtx

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clnbgrvtxel.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	clnbgrisvtx 49036	. 2 ⊢ (𝑛 ∈ (𝐺 ClNeighbVtx 𝐾) → 𝑛 ∈ 𝑉)
3	2	ssriv 3913	1 ⊢ (𝐺 ClNeighbVtx 𝐾) ⊆ 𝑉

This proof depends on syntax axioms:   = wceq 1569   ⊆ wss 3896  ‘cfv 6543  (class class class)co 7424  Vtxcvtx 29583   ClNeighbVtx cclnbgr 49024

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8005  df-2nd 8006  df-clnbgr 49025

This theorem is used by:  clnbgrlevtx  49051  clnbgrisubgrgrim  49138  clnbgrgrim  49140  isubgr3stgrlem6  49177  isubgr3stgrlem7  49178  isubgr3stgrlem8  49179  isubgr3stgr  49181  uhgrimgrlim  49193  grlicref  49218  grlicsym  49219  clnbgr3stgrgrlim  49225  clnbgr3stgrgrlic  49226